Problem 36 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{1}{\sec t-\tan t}=\sec t+\tan t $$

Step-by-Step Solution

Answer

The identity is verified: \( \frac{1}{\sec t - \tan t} = \sec t + \tan t \).

Step 1: Rewrite Trigonometric Functions

The secant and tangent functions are defined in terms of sine and cosine. Rewrite the expression using these definitions: \( \sec t = \frac{1}{\cos t} \) and \( \tan t = \frac{\sin t}{\cos t} \). This gives us:\[ \frac{1}{\frac{1}{\cos t} - \frac{\sin t}{\cos t}}.\]

Step 2: Simplify the Denominator

Combine the terms in the denominator, since they have the same denominator:\[ \frac{1}{\frac{1-\sin t}{\cos t}}.\]

Step 3: Simplify Fraction Division

To simplify the fraction, multiply by the reciprocal of the denominator: \[ 1 \times \frac{\cos t}{1 - \sin t} = \frac{\cos t}{1 - \sin t}.\]

Step 4: Use Trigonometric Identity

We know that \( 1 - \sin^2 t = \cos^2 t \) from the Pythagorean identity. So, we can rewrite \( 1 - \sin t \) using this identity:\[ \frac{\cos t (1 + \sin t)}{\cos^2 t} = \frac{\cos t(1 + \sin t)}{\cos t \cdot \cos t}. \]

Step 5: Cancel Common Factors

Cancel the common factor \( \cos t \) from the numerator and the denominator:\[ \frac{1 + \sin t}{\cos t}. \]

Step 6: Verify Result

Since \( \sec t = \frac{1}{\cos t} \) and \( \tan t = \frac{\sin t}{\cos t} \), we can see that:\[ \frac{1 + \sin t}{\cos t} = \frac{1}{\cos t} + \frac{\sin t}{\cos t} = \sec t + \tan t. \] Thus, the original identity is verified.

Key Concepts

Secant function

The secant function is one of the six fundamental trigonometric functions. It is denoted as \( \sec t \). This function is the reciprocal of the cosine function. That means the secant function is defined as:

• \( \sec t = \frac{1}{\cos t} \)

This definition implies that for any angle \( t \), when the cosine is zero, the secant is undefined.
Normally, secant is less commonly introduced than sine or cosine, but it plays an important role in various trigonometric identities and equations.
To use secant in expressions, it's crucial to remember its relationship with cosine, as this will help simplify various trigonometric identities and work through equations easily.

Tangent function

The tangent function, denoted as \( \tan t \), is another primary trigonometric function. It is defined as the ratio of the sine function to the cosine function:

• \( \tan t = \frac{\sin t}{\cos t} \)

Understanding this definition is critical because it illustrates how the tangent function is closely associated with both sine and cosine functions.
Whenever you're working with trigonometric expressions or identities, using this relationship is key for simplification. This makes it possible to convert between these trigonometric functions without any difficulty.
The tangent function is particularly useful in solving problems involving right triangles and circles. Knowing the tangent's relationship with sine and cosine can also help you tackle more complex identities.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental and useful trigonometric identities. It relates the square of sine and cosine for an angle \( t \):

• \( 1 - \sin^2 t = \cos^2 t \)

This identity is derived from the geometric Pythagorean theorem, which is why it's called the Pythagorean identity.
It provides a way to switch between squares of sine and cosine in equations, thus helping in the simplification and verification of trigonometric expressions.
In problems like our given exercise, the identity is pivotal in transforming expressions to verify or prove identities successfully. Recall this identity whenever you see expressions with squares of trigonometric functions.

Simplifying Trigonometric Expressions

To simplify trigonometric expressions, the key is to use known identities and relationships between the trigonometric functions. Here's a quick guide:

• Convert functions to sine and cosine whenever convenient. This can help make expressions uniform.
• Use reciprocal identities, like secant and cosecant, to transform complex fractions to more manageable terms.
• Utilize fundamental identities such as the Pythagorean identities to simplify expressions, by eliminating squared terms or separating fractions, as shown in solving the given exercise.

By employing these strategies, trigonometric expressions become easier to work with. This approach not only aids in simplifying but also in solving and verifying trigonometric equations effectively.